Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 7.1, Problem 7.5P
(a)
To determine
Derive the second order correction in energy for the given equation.
(b)
To determine
Derive the second order correction for the ground state energy of the given equation.
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Students have asked these similar questions
Problem 1.
The Morse potential, which is often used to model interatomic forces can be written in the form
U(r) = D(1-e²²(²-))²
3
wherer is the distance between the two atomic nuclei. Determine the
"spring constant" for small displacements from equilibrium for the Morse potential.
One model that is used for the interactions between animals, including fish in a school, is that the fish have an energy of interaction that is given below by a Morse potential. The fish will attract or repel each other until they reach a distance that minimizes the function V(r). The coefficients A and a are positive numbers. Complete parts (a) through (c).
V(r)=e^-r -Ae^-ar, r>0
Find the value of r that minimizes V(r).
A particle moves in one dimension x under the influence of a potential V(x) as
sketched in the figure below. The shaded region corresponds to infinite V, i.e., the
particle is not allowed to penetrate there.
V(x)
a
b
a²Vo =
If there is an energy eigenvalue E = 0, then a and V, are related by
a²Vo =
(n + ² ) ² n²
2m
3-1
n²π²
2m
a²V₁ =
(n + ²) π ²
2m
-Vo
nπ²
0
a
X
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
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Similar questions
- One model that is used for the interactions between animals, including fish in a school, is that the fish have an energy of interaction that is given below by a Morse potential. The fish will attract or repel each other until they reach a distance that minimizes the function V(r). The coefficients A and a are positive numbers. Complete parts (a) through (c). V(r)=e^-r -Ae^-ar, r>0 a) assume initially that a=1/2 and A=1 what is the behavior of V(r) as r approaches 0arrow_forwardConsider the "step" potential: V(x) = (a) Calculate the reflection coefficient, for the case E 0. (b) Calculate the reflection coefficient for the case E > Vo. (c) For a potential such as this, which does not go back to zero to the right of the barrier, the transmission coefficient is not simply |F12/A2 (with A the -Vo AV(x) Scattering from a "cliff" incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed. Show that T = E-Vo F1² E |A|² X for E> Vo. Hint: You can figure it out using Equation gantly, but less informatively-from the probability current ( What is T, for E Vo, calculate the transmission coefficient for the step potential, and check that T + R = 1.arrow_forwarda) A particle is placed in the well with an energy E< Uo. Sketch the first three energy levels, and make sure to label each one. V(x) U. +2(x) L X Xarrow_forward
- PROBLEM 3. Using the variational method, calculate the ground state en- ergy Eo of a particle in the triangular potential: U(r) = 0 r 0. Use the trial function v(x) = Cx exp(-ar), where a is a variational parameter and C is a normalization constant to be found. Compare your result for Eo with the exact solution, Eo 1.856(h? F/m)/3.arrow_forwardQ4. Given three agents with states x¡, i = (1,2,3). The agents are connected together and they update their states in continuous domain based on the following rule: 3 š¼(t) = 2 (*;(t) – x;(t). j=1 a) Represent the state progress as a linear system i = Ax, where x = [x1, X2, X3]" and with the initial condition x(0) = [x1(0), x2(0), x3(0)]'. b) Analyze the properties of A, i.e., find its eigenvalues, eigenvectors, determinant, etc. c) Find the Jordan normal form of the matrix A. Find e4t analytically. Write down the analytical solution of x(t) and calculate lim x(t). Analyze behavior of the system as time goes to infinity. t→∞ How would the state progress with time?arrow_forwardConsider the three-dimensional harmonic oscillator, for which the potential is V ( r ) = 1/2 m ω2 r2 (a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer: En = ( n + 3/2 ) ħ ω (b) Determine the degeneracy d ( n ) of Enarrow_forward
- Write down the equations and the associated boundary conditions for solving particle in a 1-D box of dimension L with a finite potential well, i.e., the potential energy U is zero inside the box, but finite outside the box. Specifically, U = U₁ for x L. Assuming that particle's energy E is less than U, what form do the solutions take? Without solving the problem (feel free to give it a try though), qualitatively compare with the case with infinitely hard walls by sketching the differences in wave functions and probability densities and describing the changes in particle momenta and energy levels (e.g., increasing or decreasing and why), for a given quantum number.arrow_forwardProblem 7. 1. Calculate the energy of a particle subject to the potential V(x) = Vo + câ?/2 if the particle is in the third excited state. 2. Calculate the energy eigenvalues for a particle moving in the potential V(x) = câ2/2+ bx. %3!arrow_forwardPROBLEM 2. Consider a spherical potential well of radius R and depth Uo, so that the potential is U(r) = -Uo at r R. Calculate the minimum value of Uc for which the well can trap a particle with l = 0. This means that SE at Uo > Uc has at least one bound ground state at l = 0 and E < 0. At Ug = Uc the bound state disappears.arrow_forward
- b): Consider two identical linear oscillators' having a spring constant k. The interaction potential is H = Ax|X2, where xi and x2 are the coordinates of the oscillators. Obtain the energy eigen values.arrow_forwardProblem 3: Chemical potential of an Einstein solid. Consider an Einstein solid for which both N and q are much greater than 1. Think of each ocillator as a separate “particle". a) Show that the chemical potential is H = -kT In (**e) b) Discuss this result in the limits N » q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?arrow_forwardone-dimensional A one-particle, system has the potential energy function V = V₁ for 0 ≤ x ≤ 1 and V = ∞ elsewhere (where Vo is a constant). a) Use the variation function = sin() for 0 ≤ x ≤ 1 and = 0 elsewhere to estimate the ground-state energy of this system. b) Calculate the % relative error.arrow_forward
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